On Codes over the Rings Fq + uFq + vFq + uvFq

Abstract

Codes over finite rings have been studied in the early 1970’s [4]. A great deal of attention has been given to codes over finite rings from 1990 [20], because of their new role in algebraic coding theory and their successful applications. In previous studies, some authors determined the structure of the ring F2 + uF2 + vF2 + uvF2, where u2 = v2 = 0 and uv = vu, also they obtained linear codes, self dual codes, cyclic codes, and consta-cyclic codes over this ring as in [24],[25],[26],[27]. In this thesis, we aim to generalize the previous studies from the ring F2+uF2+vF2+uvF2 to the ring Fq + uFq + vFq + uvFq, where q is a power of the prime p and u2 = v2 = 0 uv = vu, so we obtain the linear codes over R = Fq + uFq + vFq + uvFq, then we investigate self dual codes over R and we find that it can be generalized only when q is a power of the prime 2, also we obtain cyclic and consta-cyclic codes over R, and we generalize the Gray map used for codes over F2 + uF2 + vF2 + uvF2, finally we obtain another gray map for codes over R.